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Average properties of random walks on Galton-Watson trees. (English) Zbl 0894.60065
Summary: We study the $$\lambda$$-biased random walk on Galton-Watson trees by the Dirichlet principle and a formula of mean exit time of a Markov chain. We prove that the average of escaping probability and mean exit time are bounded by the counterparts of the corresponding random walks on $$\{0,1,2,\dots\}$$. In particular we partially verify the recent conjecture of R. Lyons, R. Pemantle and Y. Peres on the upper bound of the speed of $$\lambda$$-biased random walk on Galton-Watson trees [Probab. Theory Relat. Fields 106, No. 2, 249-264 (1996; Zbl 0859.60076)].

##### MSC:
 60G50 Sums of independent random variables; random walks 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 05C80 Random graphs (graph-theoretic aspects)
##### Keywords:
Galton-Watson trees; Dirichlet principle; random walks
Zbl 0859.60076
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